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Properties

Label 2-418-11.4-c1-0-0
Degree $2$
Conductor $418$
Sign $-0.924 - 0.382i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.670 + 2.06i)3-s + (0.309 + 0.951i)4-s + (−1.44 + 1.05i)5-s + (1.75 − 1.27i)6-s + (1.45 + 4.48i)7-s + (0.309 − 0.951i)8-s + (−1.38 − 1.00i)9-s + 1.79·10-s + (3.03 + 1.33i)11-s − 2.16·12-s + (−5.54 − 4.03i)13-s + (1.45 − 4.48i)14-s + (−1.20 − 3.69i)15-s + (−0.809 + 0.587i)16-s + (0.0745 − 0.0541i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.387 + 1.19i)3-s + (0.154 + 0.475i)4-s + (−0.647 + 0.470i)5-s + (0.716 − 0.520i)6-s + (0.550 + 1.69i)7-s + (0.109 − 0.336i)8-s + (−0.460 − 0.334i)9-s + 0.566·10-s + (0.914 + 0.403i)11-s − 0.626·12-s + (−1.53 − 1.11i)13-s + (0.389 − 1.19i)14-s + (−0.309 − 0.954i)15-s + (−0.202 + 0.146i)16-s + (0.0180 − 0.0131i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $-0.924 - 0.382i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ -0.924 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128464 + 0.646909i\)
\(L(\frac12)\) \(\approx\) \(0.128464 + 0.646909i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-3.03 - 1.33i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.670 - 2.06i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.44 - 1.05i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.45 - 4.48i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (5.54 + 4.03i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.0745 + 0.0541i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 - 4.91T + 23T^{2} \)
29 \( 1 + (0.990 + 3.04i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.76 + 4.18i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.82 - 5.62i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.11 + 3.42i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.91T + 43T^{2} \)
47 \( 1 + (-0.859 + 2.64i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.88 - 4.27i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.52 - 7.76i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-9.80 + 7.12i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + (-8.70 + 6.32i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.81 - 5.58i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-11.3 - 8.23i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.62 + 1.17i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 9.13T + 89T^{2} \)
97 \( 1 + (-9.89 - 7.18i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.56323984242292590029002018132, −10.69624194888472821238784812675, −9.732285748858606256494822728786, −9.209497034870838822678367968675, −8.132090325140168509517769281673, −7.16409017093666046936157603615, −5.60109300403190463584292770725, −4.81090817222512019445657490247, −3.53678026573608550482343062761, −2.32562943896656614037013572181, 0.55152388620646719742063852815, 1.65655374517228711584008292757, 4.01535806689723646481991688566, 5.00666397095123622606784919551, 6.68789805166554396141955020509, 7.09691236814484380958999105664, 7.71143341713087693819621183874, 8.781948368710783959928210910928, 9.814145126805838331505872385528, 11.01559752089775995539758335371

Graph of the $Z$-function along the critical line